# The Library

### PTAS for k-tour cover problem on the plane for moderately large values of k

Tools

Adamaszek, Anna, Czumaj, Artur and Lingas, Andrzej.
(2010)
*PTAS for k-tour cover problem on the plane for moderately large values of k.*
International Journal of Foundations of Computer Science, Volume 21
(Number 6).
pp. 893-904.
ISSN 0129-0541

**Full text not available from this repository.**

Official URL: http://dx.doi.org/10.1142/S0129054110007623

## Abstract

Let P be a set of n points in the Euclidean plane and let O be the origin point in the plane. In the k-tour cover problem (called frequently the capacitated vehicle routing problem), the goal is to minimize the total length of tours that cover all points in P, such that each tour starts and ends in O and covers at most k points from P. The k-tour cover problem is known to be NP-hard. It is also known to admit constant factor approximation algorithms for all values of k and even a polynomial-time approximation scheme (PTAS) for small values of k, k = circle divide (log n/ log log n). In this paper, we significantly enlarge the set of values of k for which a PTAS is provable. We present a new PTAS for all values of k <= 2(log delta n), where delta = delta(epsilon). The main technical result proved in the paper is a novel reduction of the k-tour cover problem with a set of n points to a small set of instances of the problem, each with circle divide((k/epsilon)(circle divide(1))) points.

Item Type: | Journal Article |
---|---|

Subjects: | Q Science > QA Mathematics Q Science > QA Mathematics > QA76 Electronic computers. Computer science. Computer software |

Divisions: | Faculty of Science > Computer Science |

Library of Congress Subject Headings (LCSH): | Transportation problems (Programming), Approximation algorithms |

Journal or Publication Title: | International Journal of Foundations of Computer Science |

Publisher: | World Scientific Publishing Co. Pte. Ltd. |

ISSN: | 0129-0541 |

Date: | December 2010 |

Volume: | Volume 21 |

Number: | Number 6 |

Page Range: | pp. 893-904 |

Identification Number: | 10.1142/S0129054110007623 |

Status: | Peer Reviewed |

Publication Status: | Published |

Access rights to Published version: | Restricted or Subscription Access |

Funder: | Engineering and Physical Sciences Research Council (EPSRC), Sweden. Vetenskapsrådet [Research Council], University of Warwick. Centre for Discrete Mathematics and Its Applications |

Grant number: | EP/D063191/1 (EPSRC), 621-2005-408 (VR) |

Version or Related Resource: | Adamaszek, Anna, et al. (2009). PTAS for k-tour cover problem on the plane for moderately large values of k. Lecture Notes in Computer Science, 5878, pp. 994-1003. http://wrap.warwick.ac.uk/id/eprint/5481 |

Related URLs: | |

References: | [1] S. Arora. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. Journal of the ACM, 45(5):753{782, 1998. [2] T. Asano, N. Katoh, H. Tamaki, and T. Tokuyama. Covering points in the plane by k-tours: a polynomial time approximation scheme for �xed k. IBM Tokyo Research Laboratory Research Report RT0162, 1996. [3] T. Asano, N. Katoh, H. Tamaki, and T. Tokuyama. Covering points in the plane by k-tours: Towards a polynomial time approximation scheme for general k. Proceedings of the 29th Annual ACM Symposium on Theory of Computing (STOC), pp. 275{283, 1997. [4] B.S. Baker. Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM, 41(1):153{180, 1994. [5] G. B. Dantzig and R, H. Ramser. The truck dispatching problem. Management Sci- ence, 6(1):80{91, October 1959. [6] A. Das and C. Mathieu. A quasi-polynomial time approximation scheme for Euclidean capacitated vehicle routing. Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 390{403, 2010. [7] M. L. Fischer. Vehicle routing. In Network Routing, Handbooks in Operations Re- search and Management Science, Vol. 8, edited by M. O. Ball, T. L. Magnanti, C. L. Monma, and G. L. Nemhauser, Elsevier, Amsterdam, pp. 1{33, 1995. [8] M.R. Garey and D.S. Johnson. Computers and Intractability. A Guide to the Theory of NP-completeness. W.H. Freeman and Company, New York 1979. [9] B. Golden, S. Raghavan, and E.Wasil. The Vehicle Routing Problem: Latest Advances and New Challenges. Springer, New York, 2008. [10] M. Haimovich and A.H.G. Rinnooy Kan. Bounds and heuristics for capacitated rout- ing problems. Mathematics of Operation Research, 10(4):527{542, 1985. [11] G. Laporte. The vehicle routing problem: An overview of exact and approximate algorithms. European Journal of Operational Research, 59(3):345{358, 1992. [12] J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: A sim- ple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM Journal on Computing, 28(4):1298{1309, August 1999. [13] F. Preparata and M. Shamos. Computational Geometry { an Introduction. Springer Verlag, New York, NY, 1985. [14] P. Toth and D. Vigo. The Vehicle Routing Problem. SIAM, Philadelphia, 2001. |

URI: | http://wrap.warwick.ac.uk/id/eprint/41591 |

Data sourced from Thomson Reuters' Web of Knowledge

### Actions (login required)

View Item |